The Extreme Szeged Index And Edge Szeged Index Of Two Classes Graphs

A unicyclic graph contain a unique cycle,if a unicyclic graph G has n ver-tices,then it has n edges. A polyomino system is a finite 2-connected plane graphsuch that each interior face (or say a cell) is surrounded by a regular square of lengthone. In other words,it is an edge-connected union of cells in the planar square lattice.A polyomino chain is a polyomino system,in which the joining of the centres of itsadjacent regular forms a path c₁c₂···c_n,where c_i is the centre of the i-th square.Topological indices and graph invariants based on the distances between the ver-tices of a graph are widely used in theoretical chemistry to establish relations betweenthe structure and the properties of molecules. They provide correlations with phys-ical,chemical and thermodynamic parameters of chemical compounds. The Szegedindex and edge-Szeged index are two important topological indices in theoretical chem-istry,we study the edge-Szeged index of unicyclic graphs,also study the Szeged indexand edge-Szeged index of polyomino chains.This thesis consists of three chapters. In the first chapter,we introduce somedefinitions and notations about graphs, Szeged index and edge-Szeged index. Thenwe summarize recent main results and investigation progresses about some topologicalindices. In the end of this chapter,the main results of this thesis are listed.In Chapter 2,we study the edge-Szeged index of unicyclic graphs and extremalgraphs,a sharp lower bound of the edge-Szeged index of unicyclic graphs which haven vertices and m edges are obtained,and the corresponding extremal graph is char-acterized. As a corollary of the result,a sharp lower bound of the edge-Szeged indexof unicyclic graphs which have n vertices is obtained,and the corresponding extremalgraph is characterized.In Chapter 3,we study the Szeged index and edge-Szeged index of polyominochains and extremal graphs,a sharp upper and lower bounds of the Szeged indexof polyomino chains are obtained,and the corresponding extremal graphs are con-structed. Moreover,a sharp upper bound of the edge-Szeged index of polyominochains is obtained,and the corresponding extremal graph is characterized.